You say you've done this using a mini-dp to HDMI adaptor? AFAIK this adaptor is used to connect a device with a mini-displayport OUT to a display with HDMI IN. E.g. connecting a MacBook to a TV.

Don't wish to doubt you but I don't believe it's possible to use this sort of cable to connect an XBOX360 to an iMac. Happy to be corrected. In fact I hope you're right as it's a much cheaper solution than the one I'm aware of.

To my knowledge to do what the OP was asking requires:
- a 27" iMac pre-thunderbolt(only the 27" models have mini displayport IN)
- a convertor to convert HDMI to displayport (like this or this)

External hard disk acquisition addict - but admitting the problem is the first step to a robust backupPlease use the reputation system if you think you've been helped - top right of this post